ПСЕВДОГИПЕРБОЛАЛЫҚ ТЕҢДЕУ ҮШІН КЕРІ ЕСЕПТІҢ ШЕШІМДІЛІГІ
DOI:
https://doi.org/10.26577/JMMCS.2022.v115.i3.01Кілттік сөздер:
Псевдогиперболалық теңдеу, керi есеп, Клейн-Гордон теңдеуi, Галеркин әдiсi, компакт әдiсi, шешiмнiң бар болуы және жалғыздығыАннотация
Мақалада Клейн-Гордон теңдеуі деген атпен белгілі псевдогиперболалық теңдеудің шешімін және оң жақ коэффициентін табукері есебізерттеледі. Бұл есеп ізделінді коэффициенттің тек уақыттан тәуелді функция болуымен ерекшеленеді. Есеп цилиндрлік аймақта қарастырылады, әдеттегідей бастапқы-шектік есептің шарттары қойылады. Қосымша шарт ретінде интегралдық түрдегі артық анықталған шартберілген. Бұл жұмыста кері есеп жүктелген сызықтық емес псевдогиперболалық теңдеу үшін қойылған эквивалентті есепке келтіріледі.Мұндай теңдеулер уақыт бойынша ең жоғары туындыға қатысты шешілмеген дербес туындылы дифференциалдық теңдеулер класына жатады және оларды құрама типті теңдеулер деп те атайды. Дәлелдеуде Галеркин әдісі және компакт әдісі (априорлық бағалаулар алу арқылы) қолданылады. Жұмыста зерттеліп отырған есептің сәйкес кластардағышешімнің бар болу және жалғыздық теоремаларыдәлелденеді.
Библиографиялық сілтемелер
[1] Greiner W., Relativistic Quantum Mechanics - Wave Equations, 3rd ed. Berlin: Springer; 2000.
[2] Dodd R. K. Solitons and nonlinear wave equations / R. K. Dodd, I. C. Eilbeck, J. D. Gibbon,H. C. Morris. L.: Acad.
Press, 1982.
[3] Strauss W, Vazquez L. Numerical solution of a nonlinear Klein-Gordon equation.JComput Phys. 1978;28(2):271-278.
[4] Jim nez S, Vazquez L. Analysis of four numerical schemes for a nonlinear Klein-Gordon equation.Appl Math Comput. 1990;35(1):61-94.
[5] Wong YS, Qianshun C, Lianger G. An initial-boundary value problem of a nonlinear Klein-Gordon equation.Appl Math Comput.1997;84(1):77-93.
[6] Wazwaz A. M. Compactons, solitons and periodic solutions for some forms of nonlinear Klein-Gordon equations // Chaos Solitons Fractals. 2006. V. 28. P. 1005–1013.
[7] Sassaman R., Biswas A. Soliton perturbation theory four phi-four model and nonlinear Klein-Gordon equations // Comm. Nonlinear Sci. Numer. Simulat. 2009. V. 14. P. 3239–3249.
[8] Sassaman R., Biswas A. Topological and non-topological solitons of the generalized Klein-Gordon equations // Appl. Math. Comput. 2009. V. 215. P. 212–220.
[9] Bratsos A. G., Petrakis L. A. A modified predictor-corrector scheme for the Klein-Gordonequation // Intern. J. Comput. Math. 2010. V. 87, N 8. P. 1892–1904.
[10] B¨ohme C, Michael R. A scale-invariant Klein-Gordon model with time-dependent potential.Ann Univ Ferrara.
2012;58(2):229-250. [11] Bhme C, Michael R. Energy bounds for Klein-Gordon equations with time-dependent potential.Ann Univ Ferrara. 2013;59(1):31-55.
[12] Hao Cheng, Xiyu Mu, GreenTMs function for the boundary value problem of the static Klein-Gordon equation stated on a rectangular region and its convergence analysis. Boundary Value Problems (2017) 2017:72.
[13] Tekin I, Mehraliyev YT, Ismailov MI. Existence and uniqueness of an inverse problemfor nonlinear Klein-Gordon equation.Math Meth Appl Sci. 2019;1-15.https://doi.org/10.1002/mma.5609
[14] Sobolev, S. L.; On certain new problem of mathematical physics. Izvestia Acad. Nauk.Mathem. 1954, no. 18, p. 3-50.
[15] Kozhanov, A.I., Comparison Theorems and Solvability of Boundary Value Problems for Some Classes of Evolution Equations Like Pseudoparabolic and Pseudohyperbolic Ones, Preprint Inst. Mat. Acad. Sci., Novosibirsk, 1990, no. 17
[16] Demidenko, G., The Cauchy problem for pseudohyperbolic equations, Selcuk J Appl Math, Vol. 1,No.1, pp.47-62, 2000.
[17] Sveshnikov, A. G.; Alshin, A. B.; Korpusov, O. M.; Pletner, Yu. D.; Linear and nonlinearSobolev type equations.
Moscow.,Phizmatlit. 2007.
[18] Korpusov, O. M., Blow-up in nonclassical wave equations. Moscow, URSS. 2010.
[19] Pulkina, L.S. Solution to nonlocal problems of pseudohyperbolic equations // Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 116, pp. 1–9.
[20] Yamamoto, M., Zhang, X.: Global uniqueness and stability for a class of multidimensional inverse hyperbolic problems with two unknowns. Appl. Math. Optim. 48, 211–228 (2003)
[21] Oussaeif,T.E., Bouziani, A.: Inverse problem of hyperbolic equationwith an integral overdetermination condition. Electron J. Diff. Equ. 2016(138), 1–7 (2016)
[22] Shahrouzi, M.: On behavior of solutions to a class of nonlinear hyperbolic inverse source problem. Acta. Math. Sin-English Ser. 32(6), 683–698 (2016)
[23] Shahrouzi, M.: Blow up of solutions to a class of damped viscoelastic inverse source problem. Differ. Equ. Dyn. Syst. 28, 889-899 (2020)
[24] Shahrouzi, M.: General decay and blow up of solutions for a class of inverse problem with elasticity term and variableexponent nonlinearities. Math Meth Appl Sci, (2021), Early view, https://doi.org/10.1002/mma.789
[2] Dodd R. K. Solitons and nonlinear wave equations / R. K. Dodd, I. C. Eilbeck, J. D. Gibbon,H. C. Morris. L.: Acad.
Press, 1982.
[3] Strauss W, Vazquez L. Numerical solution of a nonlinear Klein-Gordon equation.JComput Phys. 1978;28(2):271-278.
[4] Jim nez S, Vazquez L. Analysis of four numerical schemes for a nonlinear Klein-Gordon equation.Appl Math Comput. 1990;35(1):61-94.
[5] Wong YS, Qianshun C, Lianger G. An initial-boundary value problem of a nonlinear Klein-Gordon equation.Appl Math Comput.1997;84(1):77-93.
[6] Wazwaz A. M. Compactons, solitons and periodic solutions for some forms of nonlinear Klein-Gordon equations // Chaos Solitons Fractals. 2006. V. 28. P. 1005–1013.
[7] Sassaman R., Biswas A. Soliton perturbation theory four phi-four model and nonlinear Klein-Gordon equations // Comm. Nonlinear Sci. Numer. Simulat. 2009. V. 14. P. 3239–3249.
[8] Sassaman R., Biswas A. Topological and non-topological solitons of the generalized Klein-Gordon equations // Appl. Math. Comput. 2009. V. 215. P. 212–220.
[9] Bratsos A. G., Petrakis L. A. A modified predictor-corrector scheme for the Klein-Gordonequation // Intern. J. Comput. Math. 2010. V. 87, N 8. P. 1892–1904.
[10] B¨ohme C, Michael R. A scale-invariant Klein-Gordon model with time-dependent potential.Ann Univ Ferrara.
2012;58(2):229-250. [11] Bhme C, Michael R. Energy bounds for Klein-Gordon equations with time-dependent potential.Ann Univ Ferrara. 2013;59(1):31-55.
[12] Hao Cheng, Xiyu Mu, GreenTMs function for the boundary value problem of the static Klein-Gordon equation stated on a rectangular region and its convergence analysis. Boundary Value Problems (2017) 2017:72.
[13] Tekin I, Mehraliyev YT, Ismailov MI. Existence and uniqueness of an inverse problemfor nonlinear Klein-Gordon equation.Math Meth Appl Sci. 2019;1-15.https://doi.org/10.1002/mma.5609
[14] Sobolev, S. L.; On certain new problem of mathematical physics. Izvestia Acad. Nauk.Mathem. 1954, no. 18, p. 3-50.
[15] Kozhanov, A.I., Comparison Theorems and Solvability of Boundary Value Problems for Some Classes of Evolution Equations Like Pseudoparabolic and Pseudohyperbolic Ones, Preprint Inst. Mat. Acad. Sci., Novosibirsk, 1990, no. 17
[16] Demidenko, G., The Cauchy problem for pseudohyperbolic equations, Selcuk J Appl Math, Vol. 1,No.1, pp.47-62, 2000.
[17] Sveshnikov, A. G.; Alshin, A. B.; Korpusov, O. M.; Pletner, Yu. D.; Linear and nonlinearSobolev type equations.
Moscow.,Phizmatlit. 2007.
[18] Korpusov, O. M., Blow-up in nonclassical wave equations. Moscow, URSS. 2010.
[19] Pulkina, L.S. Solution to nonlocal problems of pseudohyperbolic equations // Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 116, pp. 1–9.
[20] Yamamoto, M., Zhang, X.: Global uniqueness and stability for a class of multidimensional inverse hyperbolic problems with two unknowns. Appl. Math. Optim. 48, 211–228 (2003)
[21] Oussaeif,T.E., Bouziani, A.: Inverse problem of hyperbolic equationwith an integral overdetermination condition. Electron J. Diff. Equ. 2016(138), 1–7 (2016)
[22] Shahrouzi, M.: On behavior of solutions to a class of nonlinear hyperbolic inverse source problem. Acta. Math. Sin-English Ser. 32(6), 683–698 (2016)
[23] Shahrouzi, M.: Blow up of solutions to a class of damped viscoelastic inverse source problem. Differ. Equ. Dyn. Syst. 28, 889-899 (2020)
[24] Shahrouzi, M.: General decay and blow up of solutions for a class of inverse problem with elasticity term and variableexponent nonlinearities. Math Meth Appl Sci, (2021), Early view, https://doi.org/10.1002/mma.789
Жүктелулер
Как цитировать
Aitzhanov, S., Ferreira, J., & Zhalgassova, K. (2022). ПСЕВДОГИПЕРБОЛАЛЫҚ ТЕҢДЕУ ҮШІН КЕРІ ЕСЕПТІҢ ШЕШІМДІЛІГІ. Қазұу Хабаршысы. Математика, механика, информатика сериясы, 115(3), 3–15. https://doi.org/10.26577/JMMCS.2022.v115.i3.01
Шығарылым
Бөлім
Математика
